Printing a game model that looks rendered
This looks like a perfect time to use a Subdivision Surface, which is a modifier that adds additional polygons for each polygon in your base mesh.  The default implementation of this also smooths them out, which should produce the result you're looking for.  

You can then apply the modifier and turn it into real geometry, and then send that geometry to the 3d printer.

The easiest way to apply a subdivision surface is select the object and press ctrl-2, which (at least in the default keymap) is the shortcut for apply a subdivision surface modifier with a 2x resolution, effectively subdivide-smoothing the mesh twice over.  Once it's on the modifier stack, you can adjust the number of subdivisions until it looks how you want.

It might make any creased areas of the mesh overly smooth though - I can't tell from the original image how sharp it's intended to look, but there are ways around that problem too - investigate the bevel modifier with the limit method set to angle, and experiment with the angle value.  30 or 60 degrees usually looks good and preserves some but not all creases.  Make sure to put the bevel above and thus before the subdivision modifier, or it won't appear to do anything.

Here's a couple of screenshots with a default suzanne, then a subdivision surface suzanne, and then a bevel at 60 degreees + subdivision suzanne:

 - base suzanne mesh.

 - with the subdisivion modifier

 - also with the bevel at 60 degrees or higher.  Note the eyebrow creases.